Basic Sine and Cosine Waves
A sine wave is the graph of a sine function.
y=sin x
Consider the table of values:
The domain of y=sin x is [-∞,∞].
The range is [-1,1]. This makes the amplitude of y=sin x 1.
A cosine wave is the graph of a cosine function.
y=cos x
Consider the table of values:
The graph of y=cos x completes one cycle from [0,2π]. This makes the period of y=cos x 2π.
The domain of y=cos x is [-∞,∞].
The range is [-1,1]. This makes the amplitude of y=cos x 1.
Transformations of Sine and Cosine Functions
Consider the functions y = a sin[b(x-c)] + d and y = a cos[b(x-c)] + d where a, b, c, and d are all constants. These constants affect the graphs of the parent functions y=sin x and y=cos x in the following ways:
a = vertical stretch factor
b = horizontal stretch factor
c = horizontal shift (phase shift)
d = vertical shift (midline shift)
Amplitude and Period of Sine and Cosine Waves
The amplitude of y=sin x and y=cos x represents half the distance between the maximum and minimum values of a function and is given by
Amplitude = |a|
The period of y=sin x and y=cos x is the time it takes to complete one cycle of the unit circle and is given by
Period = 2π / |b|
Even and Odd Functions
Sine is an odd function:
A function f is odd if, for each x in the domain of f,
f(-x) = -f(x)
To prove this,
sin(-π/2) = -sin(π/2)
-1 = -(1)
-1 = -1
Also, the graph of y=sin x is symmetric about the origin.
Cosine is an even function.
A function f is even if, for each x in the domain of f,
f(-x) = f(x)
To prove this,
cos(-π) = cos(π)
-1 = -1
Also, the graph of y=cos x is symmetric about the y-axis.
